# Properties

 Label 243.2.c.c Level $243$ Weight $2$ Character orbit 243.c Analytic conductor $1.940$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,2,Mod(82,243)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("243.82");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$243 = 3^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 243.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.94036476912$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (4 \beta_1 - 4) q^{4} + (\beta_{3} - \beta_{2}) q^{5} - 2 \beta_1 q^{7} + 2 \beta_{3} q^{8}+O(q^{10})$$ q - b2 * q^2 + (4*b1 - 4) * q^4 + (b3 - b2) * q^5 - 2*b1 * q^7 + 2*b3 * q^8 $$q - \beta_{2} q^{2} + (4 \beta_1 - 4) q^{4} + (\beta_{3} - \beta_{2}) q^{5} - 2 \beta_1 q^{7} + 2 \beta_{3} q^{8} - 6 q^{10} - \beta_{2} q^{11} + ( - \beta_1 + 1) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{14} - 4 \beta_1 q^{16} - 3 \beta_{3} q^{17} - q^{19} + 4 \beta_{2} q^{20} + (6 \beta_1 - 6) q^{22} + (\beta_{3} - \beta_{2}) q^{23} - \beta_1 q^{25} - \beta_{3} q^{26} + 8 q^{28} + 2 \beta_{2} q^{29} + ( - \beta_1 + 1) q^{31} + 18 \beta_1 q^{34} - 2 \beta_{3} q^{35} + 8 q^{37} + \beta_{2} q^{38} + ( - 12 \beta_1 + 12) q^{40} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{41} - 11 \beta_1 q^{43} + 4 \beta_{3} q^{44} - 6 q^{46} - 4 \beta_{2} q^{47} + ( - 3 \beta_1 + 3) q^{49} + ( - \beta_{3} + \beta_{2}) q^{50} + 4 \beta_1 q^{52} + 3 \beta_{3} q^{53} - 6 q^{55} - 4 \beta_{2} q^{56} + ( - 12 \beta_1 + 12) q^{58} + (\beta_{3} - \beta_{2}) q^{59} - 5 \beta_1 q^{61} - \beta_{3} q^{62} - 8 q^{64} - \beta_{2} q^{65} + ( - 7 \beta_1 + 7) q^{67} + (12 \beta_{3} - 12 \beta_{2}) q^{68} + 12 \beta_1 q^{70} + 3 \beta_{3} q^{71} + 11 q^{73} - 8 \beta_{2} q^{74} + ( - 4 \beta_1 + 4) q^{76} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{77} + 7 \beta_1 q^{79} - 4 \beta_{3} q^{80} + 12 q^{82} + 5 \beta_{2} q^{83} + (18 \beta_1 - 18) q^{85} + ( - 11 \beta_{3} + 11 \beta_{2}) q^{86} - 12 \beta_1 q^{88} - 2 q^{91} + 4 \beta_{2} q^{92} + (24 \beta_1 - 24) q^{94} + ( - \beta_{3} + \beta_{2}) q^{95} + 7 \beta_1 q^{97} - 3 \beta_{3} q^{98}+O(q^{100})$$ q - b2 * q^2 + (4*b1 - 4) * q^4 + (b3 - b2) * q^5 - 2*b1 * q^7 + 2*b3 * q^8 - 6 * q^10 - b2 * q^11 + (-b1 + 1) * q^13 + (-2*b3 + 2*b2) * q^14 - 4*b1 * q^16 - 3*b3 * q^17 - q^19 + 4*b2 * q^20 + (6*b1 - 6) * q^22 + (b3 - b2) * q^23 - b1 * q^25 - b3 * q^26 + 8 * q^28 + 2*b2 * q^29 + (-b1 + 1) * q^31 + 18*b1 * q^34 - 2*b3 * q^35 + 8 * q^37 + b2 * q^38 + (-12*b1 + 12) * q^40 + (-2*b3 + 2*b2) * q^41 - 11*b1 * q^43 + 4*b3 * q^44 - 6 * q^46 - 4*b2 * q^47 + (-3*b1 + 3) * q^49 + (-b3 + b2) * q^50 + 4*b1 * q^52 + 3*b3 * q^53 - 6 * q^55 - 4*b2 * q^56 + (-12*b1 + 12) * q^58 + (b3 - b2) * q^59 - 5*b1 * q^61 - b3 * q^62 - 8 * q^64 - b2 * q^65 + (-7*b1 + 7) * q^67 + (12*b3 - 12*b2) * q^68 + 12*b1 * q^70 + 3*b3 * q^71 + 11 * q^73 - 8*b2 * q^74 + (-4*b1 + 4) * q^76 + (-2*b3 + 2*b2) * q^77 + 7*b1 * q^79 - 4*b3 * q^80 + 12 * q^82 + 5*b2 * q^83 + (18*b1 - 18) * q^85 + (-11*b3 + 11*b2) * q^86 - 12*b1 * q^88 - 2 * q^91 + 4*b2 * q^92 + (24*b1 - 24) * q^94 + (-b3 + b2) * q^95 + 7*b1 * q^97 - 3*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} - 4 q^{7}+O(q^{10})$$ 4 * q - 8 * q^4 - 4 * q^7 $$4 q - 8 q^{4} - 4 q^{7} - 24 q^{10} + 2 q^{13} - 8 q^{16} - 4 q^{19} - 12 q^{22} - 2 q^{25} + 32 q^{28} + 2 q^{31} + 36 q^{34} + 32 q^{37} + 24 q^{40} - 22 q^{43} - 24 q^{46} + 6 q^{49} + 8 q^{52} - 24 q^{55} + 24 q^{58} - 10 q^{61} - 32 q^{64} + 14 q^{67} + 24 q^{70} + 44 q^{73} + 8 q^{76} + 14 q^{79} + 48 q^{82} - 36 q^{85} - 24 q^{88} - 8 q^{91} - 48 q^{94} + 14 q^{97}+O(q^{100})$$ 4 * q - 8 * q^4 - 4 * q^7 - 24 * q^10 + 2 * q^13 - 8 * q^16 - 4 * q^19 - 12 * q^22 - 2 * q^25 + 32 * q^28 + 2 * q^31 + 36 * q^34 + 32 * q^37 + 24 * q^40 - 22 * q^43 - 24 * q^46 + 6 * q^49 + 8 * q^52 - 24 * q^55 + 24 * q^58 - 10 * q^61 - 32 * q^64 + 14 * q^67 + 24 * q^70 + 44 * q^73 + 8 * q^76 + 14 * q^79 + 48 * q^82 - 36 * q^85 - 24 * q^88 - 8 * q^91 - 48 * q^94 + 14 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 2$$ (v^3 + 2*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu ) / 2$$ (-v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 4\beta_{2} ) / 3$$ (-2*b3 + 4*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
−1.22474 2.12132i 0 −2.00000 + 3.46410i 1.22474 2.12132i 0 −1.00000 1.73205i 4.89898 0 −6.00000
82.2 1.22474 + 2.12132i 0 −2.00000 + 3.46410i −1.22474 + 2.12132i 0 −1.00000 1.73205i −4.89898 0 −6.00000
163.1 −1.22474 + 2.12132i 0 −2.00000 3.46410i 1.22474 + 2.12132i 0 −1.00000 + 1.73205i 4.89898 0 −6.00000
163.2 1.22474 2.12132i 0 −2.00000 3.46410i −1.22474 2.12132i 0 −1.00000 + 1.73205i −4.89898 0 −6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.2.c.c 4
3.b odd 2 1 inner 243.2.c.c 4
9.c even 3 1 243.2.a.d 2
9.c even 3 1 inner 243.2.c.c 4
9.d odd 6 1 243.2.a.d 2
9.d odd 6 1 inner 243.2.c.c 4
27.e even 9 6 729.2.e.p 12
27.f odd 18 6 729.2.e.p 12
36.f odd 6 1 3888.2.a.z 2
36.h even 6 1 3888.2.a.z 2
45.h odd 6 1 6075.2.a.bn 2
45.j even 6 1 6075.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.d 2 9.c even 3 1
243.2.a.d 2 9.d odd 6 1
243.2.c.c 4 1.a even 1 1 trivial
243.2.c.c 4 3.b odd 2 1 inner
243.2.c.c 4 9.c even 3 1 inner
243.2.c.c 4 9.d odd 6 1 inner
729.2.e.p 12 27.e even 9 6
729.2.e.p 12 27.f odd 18 6
3888.2.a.z 2 36.f odd 6 1
3888.2.a.z 2 36.h even 6 1
6075.2.a.bn 2 45.h odd 6 1
6075.2.a.bn 2 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(243, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 36$$ T2^4 + 6*T2^2 + 36 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 36$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6T^{2} + 36$$
$7$ $$(T^{2} + 2 T + 4)^{2}$$
$11$ $$T^{4} + 6T^{2} + 36$$
$13$ $$(T^{2} - T + 1)^{2}$$
$17$ $$(T^{2} - 54)^{2}$$
$19$ $$(T + 1)^{4}$$
$23$ $$T^{4} + 6T^{2} + 36$$
$29$ $$T^{4} + 24T^{2} + 576$$
$31$ $$(T^{2} - T + 1)^{2}$$
$37$ $$(T - 8)^{4}$$
$41$ $$T^{4} + 24T^{2} + 576$$
$43$ $$(T^{2} + 11 T + 121)^{2}$$
$47$ $$T^{4} + 96T^{2} + 9216$$
$53$ $$(T^{2} - 54)^{2}$$
$59$ $$T^{4} + 6T^{2} + 36$$
$61$ $$(T^{2} + 5 T + 25)^{2}$$
$67$ $$(T^{2} - 7 T + 49)^{2}$$
$71$ $$(T^{2} - 54)^{2}$$
$73$ $$(T - 11)^{4}$$
$79$ $$(T^{2} - 7 T + 49)^{2}$$
$83$ $$T^{4} + 150 T^{2} + 22500$$
$89$ $$T^{4}$$
$97$ $$(T^{2} - 7 T + 49)^{2}$$